Is it possible for the power factor of a L-R-C series ac circuit to be zero? Justify your answer on physical grounds.

When inductor, capacitor and resistor are connected in series with an ac power supply as shown below, the circuit is called LCR series ac circuit.

Power factor of LCR circuit is cosine of the phase angle of voltage with respect to current.

Phase angle $\mathit{\varphi }$is expressed as

$\mathit{\varphi }\mathbf{=}\mathit{a}\mathit{r}\mathit{c}\mathit{t}\mathit{a}\mathit{n}\left(\frac{\omega L-{\displaystyle \frac{1}{\omega c}}}{R}\right)$where is angular frequency,L is inductance,c is capacitance andR is resistance.

Power factor is expressed as

$$\begin{gathered} \mathbf{cos\varphi }\mathbf{=}\mathbf{cos}\left(\arctan \left(\frac{\mathrm{\omega L}-{\displaystyle \frac{1}{\mathrm{\omega c}}}}{R}\right)\right) \\ \mathbf{cos\varphi }\mathbf{=}\frac{\mathbf{R}}{\sqrt{{\mathbf{R}}^{\mathbf{2}}\mathbf{+}{\left(\mathrm{\omega L}-{\displaystyle \frac{1}{\mathrm{\omega c}}}\right)}^{\mathbf{2}}}}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\left(i\right) \end{gathered}$$

The expression of power factor from equation (i) can be zero when R =0 i.e., circuit is purely capacitive or purely inductive.

In practice, Inductor or capacitor is made up of conductors only. The conductor always have resistancedue to various factors. Hence, on physical grounds even for purely capacitive or inductive circuit has some resistance which cannot be considered zero.

Therefore, power factor of LCR circuit can be made only close to zero by not connecting resistor.